How to use the Magnus Series Convergence Test for complex matrix?

Question

I have a two by two functional complex matrix $A$ belonging to the Magnus differential equation \begin{align} Y'(x)=A(x)Y(x) \end{align} I read from wiki that a convergence test for real $A$ can be constructed as follows \begin{align} \int_{x_1}^{x_2} ||A(x)||_2 dx < \pi \end{align} where I assume for a two by two matrix \begin{align} ||A(x)||_2 = \sqrt{A_{11}^2+A_{21}^2+A_{12}^2+A_{22}^2} \end{align} However what does one use for a convergence test if $A(x)$ is complex?

Answer 1

It appears that your identification of $\left\|\cdot\right\|_{2}$ with the Frobenius norm is incorrect, as the convergence criteria holds even for infinite-dimensional Banach algebras, and there is no extension of the Frobenius norms to infinite dimensional spaces. Therefore, the proper interpretation is the maximum eigenvalue modulus. (See equation 67 of the link.)